Biomedical Engineering Reference
In-Depth Information
Figure 2.2 Schematic of kinetic energy harvester.
A typical schematic of a kinetic energy generator is shown in
Figure 2.2
. This arrangement, a spring
mass system, consists of a
proof mass
m
, a spring with constant
k
(sometimes shaped as a cantile-
ver beam), and a damper
d
(that encompasses frictional and energy
generation damping terms). The spring
mass system is represented as
mzðtÞ
1
d zðtÞ
1
kzðtÞ
52
mω
2
Y
0
sinðωtÞ
(2.1)
where
z
is the relative displacement,
is the frequency in rad/s, and
Y
0
is the vibration amplitude. The steady-state solution, as presented by
Rao (1995), for a sinusoidal-driven input function is
ω
Y
0
ω
r
d
t
ω
k
2
mω
2
sin ωt
2
tan
2
1
q
ð1
2
ω
r
Þ
2
1
ð2ζ
t
ω
r
Þ
2
zðtÞ
5
(2.2)
ω
where
r
represents the ratio of input frequency to natural resonant fre-
quency
ω
5
ω
/
ω
n
,
ω
n
is the natural resonant frequency (
ω
n
5
O
km),
ζ
r
t
is the total damping ratio (
ω
n
)),
d
t
represents the total damp-
ing, while tangent term inside the last parenthesis is the phase angle.
ζ
t
5
d
t
/(2
m
The power dissipated from the system represented in
Figure 2.2
into
the damper, from El-Hami et al. (2001), is
mζ
t
Y
0
ω
r
ω
2
½1
2
ω
r
2
1
½2ζ
t
ω
r
2
P
d
5
(2.3)
The maximum power is found when the vibration frequency matches
the natural resonant frequency (
5
1
)
. The previous expression becomes
P
d
5
1
2
ω
r
mY
0
ω
n
1
(2.4)
2ζ
t
